Empirical Quantile CLTs for Time Dependent Data

TL;DR

This paper proves central limit theorems for empirical quantile processes and related empirical processes derived from various stochastic processes, with uniform convergence over time and quantile levels, applicable to broad classes of processes.

Contribution

It establishes empirical quantile process CLTs for a wide range of stochastic processes, extending the scope of previous results to more general settings and processes.

Findings

CLTs for empirical quantile processes are established.

Uniform convergence results over time and quantile levels.

Applicable to Gaussian, Poisson, stable processes, and martingales.

Abstract

We establish empirical quantile process CLTs based on $n$ independent copies of a stochastic process $\{X_t: t \in E\}$ that are uniform in $t \in E$ and quantile levels $\alpha \in I$, where $I$ is a closed sub-interval of $(0,1)$. Typically $E=[0,T]$, or a finite product of such intervals. Also included are CLT's for the empirical process based on $\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}$ that are uniform in $t \in E, y \in R$. The process $\{X_t: t \in E\}$ may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.